Free coating of viscous and viscoelastic liquids onto a partially submerged rotating roll

Experiments are performed on the dynamics of formation of a liquid coating picked up by a cylindrical roll rotating partially submerged in a free bath. Data on coating thickness for Newtonian fluids are found to correlate according to T = H(ρg/μU)^1/2 = 0.56 for fluids whose viscosities range from 0.11 to 33 poise. Data for strongly non-Newtonian and Viscoelastic fluids (polyacrylamide solutions) can be forced to fit this correlation by defining an “equivalent coating viscosity.” It is clear that this defines a pseudo viscosity, because the “equivalent coating viscosity” is observed to increase with increasing roll speed. This suggests that strongly Viscoelastic fluids respond to the rapid deformation suddenly imposed in the dynamic meniscus near the pickup point in a distinctly elastic manner that alters the flow through the meniscus.     

Effect of thickness and viscoelastic properties of roll cover on deformable roll coating flows

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter a continuous liquid layer and to apply it to a flexible substrate. Of the two rolls that make a forward-roll coating gap, one is often covered by a layer of deformable elastomer. Liquid carried into the gap develops high enough pressure to deform the resilient roll cover. The complete understanding of the coupling between the liquid flow and roll cover deformation is vital to the optimization of this widely used and simple coating method. Most of the earlier works on deformable roll coating analyzed the action with both the lubrication approximation and the full Navier-Stokes solution of the liquid flow, and one-dimensional elastic models of the roll cover deformation. The effect of the roll cover thickness was not explored and can explain some of the discrepancy observed from the available theoretical predictions and experimental measurements from different researchers. Moreover, rubber and rubber-like materials used as roll covers do not behave purely elastically. Their responses depend to a great extent on the stress history and the temperature of the roll cover. In this work, the flow between a rigid and a deformable rotating roll was examined by solving the complete Navier-Stokes system coupled with a plane-strain elastic and viscoelastic model of the roll cover deformation. The stress at each location of the roll cover was evaluated by an integral of the deformation along the material path of the point being analyzed. The equation system was solved by the Galerkin/finite element method. Results show how thickness of the roll cover and its viscoelastic properties affect the performance of deformable roll nips.     

Slot die stripe coating of low viscous fluids

Slot die coating is applied to deposit thin and homogenous films in roll-to-roll and sheet-to-sheet applications. The critical step in operation is to choose suitable process parameters within the process window. In this work, we investigate an upper limit for stripe coatings. This maximum film thickness is characterized by stripe merging which needs to be avoided in a stable process. It is shown that the upper limit reduces the process window for stripe coatings to a major extent. As a result, stripe coatings at large coating gaps and low viscosities are only possible for relatively thick films. Explaining the upper limit, a theory of balancing the side pressure in the gap region in the cross-web direction has been developed.     

A semi-empirical model of the forward roll coating flow of newtonian fluids

A theory of the forward roll coating flow of Newtonian fluids has been developed recently by Greener and Middleman [1] utilizing the usual lubrication approximations and some very simple physical notions regarding the region in which the total flux through the nip splits into two films. These basic notions restricted the validity of their model to the simple case of two rolls of equal rotational speeds and equal radii. Their analysis resulted in an expression for the dimensionless total flux through the nip λ as a weak function of a modified capillary number N.The aim of this paper is to extend the model presented by these workers [1] by introducing to it experimental evidence of the flux distribution [2] and thus generalizing its applicability to operations involving rolls of different diameters rotating at different speeds. The results of this extended model give the dimensionless total flux λ as a function of the modified capillary number N, the speed ratio r_s, and the radius ratio r_r. This function is a weak one over the range of N, r_s, and r_r, normally encountered in forward roll coating operations and is in good agreement with experimental data [2].     

Mathematical modelling of the reverse and metering roll coating flow of Newtonian fluids

A mathematical model of the reverse and metering roll coating flow of Newtonian fluids is presented. It is based on the lubrication approximation and simple physical notions regarding the separation region. These physical notions discard the effect of surface tension on the dynamics of the system. The results of this model give the dimensionless net flux through the nip, λ_L, as a function of the speed ratio, r_s and the radius ratio r_r. This function is a weak one over the range of r_r and r_r normally encountered in reverse and metering roll coating operations and is in good agreement with experimental data[1] obtained previously.The results of this present study have also been related to the findings from a previous analysis of forward roll coating[2].     

Tri-helical gravure roll coating

A mathematical model is presented and solved for the fluid flow within the coating bead of a tri-helical gravure roll coater, operating in reverse mode. A variety of rolls etched circumferentially with grooves of different cross-sectional shape and aligned at non-zero angles of pitch are investigated. Predictions of fluid pick-out from the grooves are compared with complementary experimental data. Quantitative agreement between the two is found to be very good, showing a linear increase in pickout as a function of web-to-roll speed ratio and groove depths up to the point at which streaking, as observed experimentally, occurs and beyond which the model is no longer valid. In regions of parameter space for which there is no experimental data available for comparison purposes the model predicts that: (i) fluid pick-out decreases with increasing groove depth while the film thickness tends to increase and (ii) an increase in groove aspect ratio leads to a reduction in both pick-out and film thickness for a given land width and groove cross-sectional shape.     

Blotching in roll coating

After thermoforming, plastic parts are stacked for shipping, and these parts tend to stick together. Called nesting, sheet stock is often first coated with a silicone compound before thermoforming to prevent this. The coating usually consists of a small amount of lubricant dispersed in a majority of carrier fluid, and this fluid must then dry before reaching the sheet winder or else the coating blotches. This coupling of coating and drying to determine when to expect blotching is examined. Roll coating involves a dimensionless group called the elasticity number that governs the thickness of the coating to be dried. The drying section involves the evaporation of the coating carrier fluid, and then diffusion into the dry surrounding atmosphere. When analyzing the drying, a new dimensionless group that governs blotching is discovered, called blotchability. The result of this analysis allows practitioners to determine which operating conditions cause blotching, and how to eliminate it. Roll coating uses a deflecting rubber roll to apply vanishingly thin coatings (≤1 μm), an interesting elastohydrodynamic problem.     

A theoretical and experimental investigation of tri-helical gravure roll coating

Tri-helical, gravure roll coating, operated in reverse mode, is investigated via a complementary experimental and theoretical approach. Flow visualisation reveals the underlying flow structure within the roll-to-web transfer region and highlights when loss of stability, that is a propensity to streaking, occurs. The latter is found to be influenced in turn by the depth of the tri-helical grooves and the capillary number. Experiments show that as the web-to-roll speed ratio is increased, so too is fluid pick-out from the grooves, although the coated film thickness may decrease. Extensive data are provided for coated film thickness, pick-out and meniscus location as a function of speed ratio, for different operating parameters. A key feature of the present investigation is the formulation of a novel complementary mathematical model for the process. The base flow within the grooves is modelled as a Poisson equation in conjunction with simplified approximations for the attendant free-surface boundary conditions. The resultant equation set is written in terms of finite Fourier sine transforms and unique solutions obtained by varying the flux until a consistent pressure distribution throughout the coating bead is obtained. Results from the model are found to be in generally good agreement with their experimental counter-parts, and are seen to predict and capture all the major features of the process over the operating range explored.     

A torque-based analysis of the reverse roll coating process

In a laboratory film coater operated at industrial speed, the torque exerted by the metering rod was used to investigate the hydrodynamics in the metering nip flow. CFD simulations were also performed taking into account the deformation of the roll cover to depict the elastohydrodynamics of the metering nip flow. From the experiments, it was found that the torque increases with transfer speed, load and viscosity. When runnability problems such as spitting were observed, the torque signal decreased. Torque measurements could also predict excessive friction between rod and roll, which may damage the elastic cover. The numerical simulations showed trends similar to the experiments. Regions of positive and negative torque values demonstrated how the fluid interacts with the rod, a phenomenon apparently related to the occurrence of secondary flows in the metering nip. Finally, the numerical results also showed that the minimum nip gap becomes constant at high loads.     

Unsteady flow of viscoelastic fluids

Theoretical solutions for unsteady flow of a three constant Oldroyd fluid and a second order fluid under several different flow conditions of practical interest are obtained. The response of these fluids to suddenly applied external force is investigated in each case. Without using the stick-slip boundary condition at the wall, it is possible to show that pressure oscillation occurs with both fluids under a certain case.     

A two-scale model for discrete cell gravure roll coating

A two-dimensional model for predicting the fluid pickout and coated film thickness characteristics of a discrete cell direct gravure roll coater operating in reverse mode is derived. A novel multiscale approach is adopted for this purpose and the resulting equations solved numerically for inertia-less flow conditions. A system of stiff ordinary differential equations is found to be sufficient to capture the major gross flow features, while at the cell level the analysis is based on a finite element solution of the momentum and continuity equations. It represents the first such predictive model of its kind, with particular interest placed on the nature of both the pressure distribution and web-to-roll gap profile spanning the coating bead. The effect of key operating parameters, web-to-roll speed ratio, web-tension, wrap-angle, capillary number and cell-geometry, on the degree of fluid pickout from gravure cells and the coated film thickness is explored. Although an idealised model, the trends observed show qualitative agreement with existing experimental data collected on a small-scale gravure coating rig and point the way forward to the eventual formulation of a full three-dimensional predictive model of the process.     

Die swell in elastic and viscous fluids

Measurements of die swell have been made on an elastic liquid and a Newtonian liquid of similar viscosity (～ 10⁴ N s m^?2). The Reynolds number was about 10^?8. The Newtonian liquid has a die swell of 13.5% which was independent of shear rate. The die swell of the elastic liquid increased with shear rate and seemed to be asymptotic to the Newtonian die swell at low shear rates.     

Time-dependent flow of quasi-linear viscoelastic fluids

Transient flow behavior of an incompressible quasi-linear viscoelastic fluid under suddenly applied constant pressure as well as under a periodic pressure gradient was investigated using a three-parameter relaxation function. In the light of these solutions, the roll of viscoelastic relaxation in the overshooting of volumetric flow rate and the effect of viscoelastic parameters on the mean square velocity profile are discussed.     

Turbulent flow properties of viscoelastic fluids

The abnormally high resistance of viscoelastic fluids to sudden deformations and to stretching may be expected to cause the structure of turbulent velocity fields in these systems to differ appreciably from those of Newtonian fluids. An analysis based on these considerations is presented and supported using friction factor data for three concentrations of a water soluble polymer. Pressure losses predicted by use of this correlation are within 15% or less of the experimental values, but as all dimensionless groups which influence the results were not varied independently, the correlation developed may not be universally applicable to all fluids. Suggestions for further and more more specific analyses are made.     

Roll coating of purely viscous liquids

Experimental measurements are reported for the flow of Newtonian liquids through the nip of two co- and counter-rotating rolls of various size and speed ratios. Results are presented in a dimensionless form for the total volumetric flux through the nip and for the distribution of this flux between the two rolls. Simple correlations have been developed and these should be useful in the design and analysis of roll coating equipment.Some preliminary data are also presented for a shear thinning inelastic non-Newtonian liquid. The forms of the correlating equations developed for the Newtonian systems are still applicable but the constants in the equations are slightly modified.     

A study of viscous dissipation in the calendering of power-law fluids

A finite difference procedure was used to solve the equation of conservation of energy with the lubrication approximation. The temperature profiles due to viscous dissipation in the gap formed by a pair of equal sized rolls rotating at the same speed exhibit two maxima in the vicinity of the roll surfaces. The maximum temperatures also exhibit two local maxima and a minimum in the direction of flow. The calculated temperature distributions across and along the flow field reveal the possibility of excessive local heating which might be detrimental for certain temperature sensitive materials such as poly(vinyl chloride) and rubbers.     

A method to model web trajectory and release in forward roll coating

The trajectory of the web at the exit of a roll coating operation can influence the quality of the final coating. While methods to model the web trajectory have been given in the literature, these methods are limited in various ways. A method is proposed to describe the web trajectory and the pressure distribution in the fluid at the exit of a forward roll coater. The Reynolds lubrication equations for the fluid are coupled with the web by a force balance on web node points. The fluid pressure in the coating layer generates forces on the web. These forces deflect the web. Integration in time gives the web dynamics. The angle that the web is pulled from the nip and the tension are found to influence the pressure pulse in the divergent section of the nip to a large extent. Low tensions lead to a second pressure pulse followed by a sub-ambient or tack pressure. Pulling the web at various angles from the nip can cause the tack pressure to increase or decrease. Pressure pulses are predicted that are comparable to measurements by a laboratory device.     

Dimensional analysis of flow of viscoelastic fluids

Dimensional analysis of viscoelastic flow phenomena is analyzed in general terms, and it is shown that the method can only be used for “rheologically homologous” fluids (defined rigorously in the paper). Few generally useful results can be obtained from dimensional analysis when the condition of homologousness is met by making experiments with the same fluid for which predictive equations are sought. Other ways of meeting the condition are discussed, and some results of a general nature are obtained.     

Stability of isothermal spinning of viscoelastic fluids

The stability of isothermal spinning of viscoelastic fluids which have strain-rate dependent relaxation time has been investigated using the linear stability analysis method. The instability known as draw resonance of the system was found to be dependent upon the material functions of the fluids like fluid relaxation time and the strain-rate dependency of the relaxation time as well as upon the draw-down ratio of the process. Utilizing the fundamental physics of the system characterized by the traveling kinematic waves, we also have developed a simple, approximate method for determining this draw resonance instability; it requires only the steady state velocity solutions of the system, in contrast to the exact stability analysis method which requires solving the transient equations. The stability curves produced by this simple, fast method agree well with those by the exact stability method, proving the utility of the method. The stability of other extensional deformation processes such as film casting and film blowing can also be analyzed using the method developed in this study.